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Theorem vccl 27808
Description: Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vciOLD.1 𝐺 = (1st𝑊)
vciOLD.2 𝑆 = (2nd𝑊)
vciOLD.3 𝑋 = ran 𝐺
Assertion
Ref Expression
vccl ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Proof of Theorem vccl
StepHypRef Expression
1 vciOLD.1 . . 3 𝐺 = (1st𝑊)
2 vciOLD.2 . . 3 𝑆 = (2nd𝑊)
3 vciOLD.3 . . 3 𝑋 = ran 𝐺
41, 2, 3vcsm 27807 . 2 (𝑊 ∈ CVecOLD𝑆:(ℂ × 𝑋)⟶𝑋)
5 fovrn 7001 . 2 ((𝑆:(ℂ × 𝑋)⟶𝑋𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
64, 5syl3an1 1202 1 ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1107   = wceq 1652  wcel 2155   × cxp 5274  ran crn 5277  wf 6063  cfv 6067  (class class class)co 6841  1st c1st 7363  2nd c2nd 7364  cc 10186  CVecOLDcvc 27803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-fv 6075  df-ov 6844  df-1st 7365  df-2nd 7366  df-vc 27804
This theorem is referenced by:  vc0  27819  vcm  27821  nvscl  27871
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